Mathematics for the interested outsider

We continue with our example and show that the set function which assigns any semiclosed interval its length has various monotonicity properties.

First off, let be a finite, disjoint collection of semiclosed intervals, all of which are contained in another semiclosed interval . Then we have the inequality

Indeed, we can write , , and without loss of generality assume that . Then our hypotheses tell us that

and thus

On the other hand, if is a closed interval contained in the union of a finite number of bounded open intervals , then we have the strict inequality

We can rearrange the open intervals by picking to contain . Then if we have and we can discard all the other sets since they only increase the right hand side of the inequality. But if , we can pick some containing . Now we repeat, asking whether is greater or less than . Eventually we’ll have a finite collection of satisfying , , and . It follows that

What does this have to do with semiclosed intervals? Well, if is a countable sequence of semiclosed intervals that cover another semiclosed interval , then we have the inequality

If , then this is trivially true, so we’ll assume it isn’t, and let be a positive number with . Then we have the closed set . We can also pick any positive number and define .

Now is smaller than , and each is larger than the corresponding , and so we find that is a closed interval covered by the open intervals . But the Heine-Borel theorem says that is compact, and so we can find a finite collection of the which cover . Renumbering the open intervals, we have

and our above result tells us that

Since we can pick and to be arbitrarily small, the desired inequality follows.

About this weblog

This is mainly an expository blath, with occasional high-level excursions, humorous observations, rants, and musings. The main-line exposition should be accessible to the “Generally Interested Lay Audience”, as long as you trace the links back towards the basics. Check the sidebar for specific topics (under “Categories”).

I’m in the process of tweaking some aspects of the site to make it easier to refer back to older topics, so try to make the best of it for now.